Numerical reasoning
Syllogism questions: all, some and none, with worked answers
A syllogism question asks what must follow from its premises. A conclusion can sound sensible and still be unsupported. Work through three original examples below: an inclusion chain, a known member shared by groups, and a member excluded from another group.
First, check the question’s conventions
Treat the premises as true for the exercise and use only what they establish. In this guide and Cognik’s formal inference checker, “some” means at least one and may include all. An “all” statement alone does not establish that the first group has any members. Every example below uses that convention.
Assessment instructions may specify a different convention, including assuming that a named group exists. Check the organiser’s own guidance before transferring a shortcut to that assessment. These are general reasoning exercises, not a reproduction of a named test.
What do all, some, none and some not mean?
Read the quantifier before drawing a conclusion. It controls how much of a group the statement covers.
- All A are B: A is included in B. This does not reverse to all B are A, and alone it gives no member of A.
- Some A are B: at least one member belongs to both groups. It does not say that other A members lie outside B.
- No A are B: the groups do not overlap. Reversing the groups is safe: no B are A.
- Some A are not B: at least one member of A is outside B. This is weaker than saying that no A are B.
A repeatable three-step check
Keep the same object or group in view as you move between premises. A diagram should record what is required, not add a relationship that merely looks possible.
- Mark the direction of each inclusion and separate groups that cannot overlap.
- For some or some not, place one known member. Carry that same member through the other premises.
- Try to make each proposed conclusion false while keeping every premise true. One such counterexample is enough to show that the conclusion is not necessary.
Example 1: follow an inclusion chain
Every stamped document is filed. Every filed document is catalogued. Choose the conclusion forced by these two premises under the convention above.
| Proposed conclusion | Option |
|---|---|
| Every stamped document is catalogued. | A |
| Every catalogued document is stamped. | B |
| Some stamped documents are catalogued. | C |
| No filed document is catalogued. | D |
Example 2: keep the same known member
Every checked parcel is registered. Some urgent parcels are checked. What must follow?
| Proposed conclusion | Option |
|---|---|
| Every urgent parcel is registered. | A |
| Some urgent parcels are registered. | B |
| Some registered parcels are not checked. | C |
| No urgent parcel is registered. | D |
Example 3: carry a member outside a group
Some square tokens are amber. No amber token is striped. Which conclusion must be true?
| Proposed conclusion | Option |
|---|---|
| Every square token is amber. | A |
| No square token is striped. | B |
| Some square tokens are not striped. | C |
| Some amber tokens are striped. | D |
What to check after a wrong answer
Name the missed step: reversing all, replacing some with all, inventing a member, or losing track of which object the premises describe. Rework one original example with a new arrangement rather than memorising its answer letter.
Two separate some statements need not refer to the same member. A counterexample can place their witnesses in different parts of the shared group. The Inspector 33 guide below shows two concrete arrangements when a conclusion cannot be determined.
Practice scope and reference
The examples are original and follow the inclusion, overlap and exclusion structures in Cognik’s syllogism practice. The reference explains the modern quantifier convention used here. For an actual assessment, its own directions determine the assumptions, question format and scoring.
Put it into practice
Explore more questions and step-by-step explanations in Cognik.
